Sovereign Uncertainty

Fiscal Policy Under the Debt Crisis

September 21, 2015

Abstract

How should the fiscal policy be adjusted when the volatility of government bond

yield is time varying? In this paper, I first quantify the stochastic volatility of real

government bond yields for Germany, Italy, Spain and Portugal. I show that among

peripheral countries, volatility shock was the highest for Portugal during the debt crisis,

followed by Italy and Spain. Then, I propose a small open economy model that examines

the dynamics of government revenue and other macroeconomic variables when fiscal

policy is adjusted on the impact of real interest rate volatility shock. I find that under

high volatility, consumption tax will result in the highest revenue but at the cost of

biggest welfare loss compared to the benchmark low volatility. Capital and labor income

taxes show similar result but to the lesser amount. Calibrating to Portuguese data, I

show that 1 percentage point increase in consumption tax will yield 1.5% less total

government revenue and 1.6% more welfare loss in present value terms under the high

volatility than the low volatility.

JEL-Classification: E13, E20, E42, E62, F32, F41, H21

Keywords: debt crisis, stochastic volatility, fiscal policy, small open economy.

1

1 Introduction

Peripheral European countries implemented austerity programs while their government bond

yields became more volatile. In the wake of Great Recession in 2009, public balance of

Italy, Portugal, Spain and Greece deteriorated significantly. Spain and Greece recorded net

borrowing of 11% and 15.3% of GDP, respectively. At the same time, government bond yields

showed increasing volatility. In July 2009, Spain’s 3 year government bond yield in real terms

increased by 396 basis points compared to August 2008. As can be seen in Figure 11, real

yields for all other governments also rose by more than 200 basis points over the same period.

In the middle of economic turmoil with increasing bond yield and deepening recession, these

governments undertook austerity programs over the course of 2009-2013 (Alesina et al., 2015)

in an effort to reduce fiscal deficit and public debt.

How should the fiscal policy be adjusted when the volatility of government bond yield is time

varying? The experience of peripheral European governments described above leads one to

question the impact of fiscal policy when there is volatility shock. As recent studies show2,

volatility shocks can have significant impact on the macroeconomic variables that serve as

the tax revenue base. This implies that the same change in fiscal policy can yield different

results in terms of government revenue and household welfare depending on the volatility.

Classic studies based on the static Laffer curve, however, will prescribe same tax rates re-

gardless of the volatilities. This can be misleading when the impact of volatility is sizable.

In this paper, I first quantify the stochastic volatility of real government bond yields for four

European countries: Germany, Italy, Spain and Portugal. From the estimation using sequen-

tial Markov Chain Monte Carlo algorithm, I show that there was significant rise in volatility

in 2009-2013. Then, I propose a small open economy model that examines the dynamics

1Real government bond yield in the graph is nominal yield of 3 year maturity government bond net of

Harmonized Consumer Price Index (overall index) of each country. Greek bond yield peaked at 7595 basis

point in February 2012 and defaulted from March 2012 to June 2014, which is not shown in the graph due to

space limit. Data is sourced from central bank of each country for nominal government bond yield, Bundes

Bank for HCPI and Eurostat for government revenue to GDP.2See Bloom (2009) and Fernandez-Villaverde et al. (2011a), among others.

2

Figure 1: Real 3 Year Government Bond Yield

of government revenue and other macroeconomic variables when fiscal policy is adjusted on

the impact of real interest rate volatility shock. Calibrating to European data, I show that

static model will overestimate increase in government revenue and underestimate decrease in

household welfare compared to the dynamic model with stochastic volatility.

In the model, households face stochastic volatility in bond yields as well as taxes on consump-

tion and factor incomes. The government finances its expenditure by levying taxes on the

households and issuing domestic bonds. Households smooth their consumption by borrowing

from foreigners3.

In this environment, I first compare two economies where the real interest rate is more volatile

in one economy than the other. By increasing the tax rate together with real interest rate,

I analyze the difference in consumption, labor supply, investment, borrowing decisions of

households, as well as government revenue and household welfare across the two economies.

For the same 1% of each tax increase, I show that household decisions vary across the different

taxes and volatilities. Private agents in more volatile economy choose to run down debt level

3In this paper, it is assumed for simplicity that only household will borrow from the foreigners. There is

indeterminacy of private and public debt due to the presence of transfer. More details will be discussed in

the section 3 of the paper.

3

more, consume less and work more compared to those in less volatile one on impact of the

shock. The intuition is that higher volatility leads to more precautionary behavior, since

there is riskier bond that the agent can borrow to smooth out the negative shock. Moreover,

increase in consumption tax leads to more investment since it is relatively cheaper to save

in capital than to borrow when there is increase in real interest rate. For factor income tax

shock, on the other hand, households in both volatile and stable economies decide to decrease

the investment since factor income tax is proportional to output.

Also, I present simulated Laffer curves for both volatile and stable economies and compare

them to the static economy without any real interest rate shock. In this exercise, I fix the

tax rate and simulate the model under two different volatilities of real interest rate. I show

that higher volatility will shift the Laffer curve down across all tax rates, but the revenue is

maximized at the same tax rate. The reason why higher volatility lowers the level of revenue

is that consumption and output level will be lower in the more volatile economy. Moreover,

the Laffer curve will still peak at the same tax rate since difference in volatility mainly scales

the household decision up and down, not changing the direction of response to the shock.

This paper mainly weaves three strands of literature: uncertainty, fiscal policy and small open

economy business cycle. The model is based on the canonical small open economy model

as in Neumeyer and Perri (2005) and more recently Fernandez-Villaverde et al. (2011a),

with imperfect financial market of risk free asset and production. The main difference from

existing literature is in incorporating stochastic tax wedge to the households and focusing on

the impact of volatility on government revenue. The proposed model in this paper can be

served as a basic framework for small open economy business cycle accounting, expanding

Chari et al. (2007) which is for the closed economy without volatility shock in real interest

rate.

In both estimation and models, I follow the framework of Fernandez-Villaverde et al. (2011a),

which studies the impact of volatility shock in small open economy by introducing time

varying volatility in the real interest rate process. However, their paper does not have

government in the model, and its focus is not on the Laffer curve as in this paper.

Another study by Fernandez-Villaverde et al. (2011b) examines volatility in fiscal rule and

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its impact in the US economy, based on time varying volatility process and particle filtering.

Born and Pfeifer (2014a) investigates on uncertainty of US fiscal policy as well, but both

studies are on closed economy and fiscal rules.

Finally, regarding European fiscal policy, Mendoza et al. (2014) studies effect of austerity

programs in European countries hit by debt shock. Based on two-country model, the paper

shows externality of fiscal policy in open economies, especially on capital taxation. However,

their paper is set up in neoclassical models without shocks and mainly discuss transition to

new steady state with the debt shock and austerity programs.

The rest of paper is organized as follows. Section 2 describes European debt crisis and auster-

ity statistics in 2009-2013 and estimates stochastic volatility of real government bond yields

for four European countries using particle filtering. Then, Section 3 builds the model of small

open economy with government and volatility in real interest rate. Section 4 analyzes the

model mechanism by impulse response functions across different taxes and presents simulated

Laffer curves based on calibration. Section 5 concludes. Appendix includes more details on

model equations. Data Appendix explains the source of data used in the paper and data

description omitted in the main body of the paper.

2 Data and Estimation

In this section, I first describe public balance and government bond yield statistics for pe-

ripheral European countries: Italy, Portugal, Spain and Greece. I show that these countries

undertook fiscal adjustment while suffering from increased volatility in their government bond

yield. Then, I estimate the time varying volatility of real government bond yield. I set up real

interest rate process composed of stochastic level and volatility, and use sequential Markov

Chain Monte Carlo algorithm called particle filtering to recover the time varying volatility.

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Figure 2: General Government Balance

Figure 3: General Government Revenue to GDP

2.1 European Debt Crisis and Austerity

In 2009-2013, peripheral European countries were hit by a series of adverse shocks. Especially,

sovereign governments suffered from mounting debt and underwent fiscal adjustment while

government bond yields were increasingly volatile.

As Figure 2 shows, public balance of all four countries significantly deteriorated in the wake

of the Great Recession in 2009. Spain and Greece recorded net borrowing of 11% and 15.3%

of GDP, respectively. At the same time, government bond yields became more volatile. In

July 2009, Spain’s 3 year government bond yield in real terms increased by 396 basis points

6

compared to August 2008. As can be seen in Figure 1, real yields for all other governments

also rose by more than 200 bp over the same period.

In response, these governments implemented austerity program in 2009-2013 and targeted

to reduce their fiscal deficit (Alesina et al., 2015). However, it was precisely the period that

government bond volatilities were at their highest. Therefore, fiscal adjustment of sovereign

government was subject to significant aggregate volatility shock. The observed time series of

government revenue to GDP in Figure 3 can be interpreted as a realized equilibrium of fiscal

adjustment under the volatility shock of real interest rate 4.

Motivated by the peripheral European governments’ experience in debt crisis and austerity

program described above, this paper analyzes the impact of volatility shock on the govern-

ment revenue and macroeconomic variables based on European data. As its first step, I

estimate time varying volatility in real interest rate for four countries: Germany, Italy, Por-

tugal and Spain. In the following subsection, I describe the data and estimation procedure.

2.2 Estimation

Real Interest Rate Data I briefly explain the data used for estimation. I construct real

interest rates by subtracting Harmonized Consumer Price Index from nominal government

bond yields, 3 year maturity. Then, German real interest rate is assumed to be the risk

free rate, and country spread is derived by subtracting German rate from each country’s real

interest rate. I use monthly data from January 2000 to April 2015. Table 1 summarizes the

mean and standard deviation of the real interest rates used in the estimation. All figures are

in percent per annum. More details for the data source are in Data Appendix.

Real Interest Rate Process Real interest rate (rt) is composed of mean real interest

rate (r), risk-free rate shock (εf,t) and country spread shock (εr,t). This structure follows

4It is true that GDP decreased for these countries especially in 2011-2013. However, as Figures 13 and

14 show in Appendix, the increase in government revenue to GDP ratio is not entirely due to drop in the

GDP. Real government revenue did increase for most countries except for Greece, with Portugal increasing

its revenue to the level of 2008 in year 2011.

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Table 1: Real Interest Rate Summary Statistics

Germany Italy Portugal Spain

Mean 0.8054 1.0806 2.1843 0.7223

Sd 1.5644 0.8544 3.1493 1.0965

Fernandez-Villaverde et al. (2011a).

rt = r + εf,t + εr,t (1)

Country spread shock εr,t follows AR(1) process with level shock ur,t multiplied to the time

varying volatility σt.

εr,t = ρrεr,t−1 + exp(σr,t)ur,t (2)

Time varying volatility process σt follows AR(1) process as well, with unconditional mean σ.

Standard deviation of the volatility shock uσ,t is given as ησ.

σr,t = (1− ρσr)σr + ρσrσr,t−1 + ησruσr,t (3)

Both level and volatility shocks ur,t and uσ,t are assume to follow i.i.d. standard normal

N(0, 1), independent to each other. Risk free rate shock follows similar process. Formally

put,

εf,t = ρfεf,t−1 + exp(σf,t)uf,t (4)

σf,t = (1− ρσf )σf + ρσfσf,t−1 + ησfuσf ,t (5)

where both uf,t, uσf ,t ∼i.i.d N(0.1).

Estimation Result Estimation method is based on sequential Monte Carlo algorithm,

or particle filtering5. Throughout the estimation, 2,000 particles are used for filtering and

2,000 particles are used for backward smoothing. Burn-in periods are 5,000 and Metropolis-

Hastings draws are performed 20,000 times. In Markov Chain Monte Carlo algorithm, scale

factors are used so that proposal density has acceptance rate of 25-40%.

5Refer to Fernandez-Villaverde et al. (2011a), Born and Pfeifer (2014b) and Fernandez-Villaverde et al.

(2015) for more detailed explanation of the algorithm.

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Table 2: Prior Distribution

Germany Spain Portugal Italy

ρ B(0.9, 0.02) B(0.9, 0.02) B(0.9, 0.02) B(0.9, 0.02)

ρσ B(0.95, 0.02) B(0.9, 0.05) B(0.9, 0.05) B(0.9, 0.05)

σ lnN (−2, 0.3) lnN (−2, 0.3) lnN (−2, 0.3) lnN (−2, 0.3)

η Γ(0.2, 0.05) Γ(0.3, 0.05) Γ(0.3, 0.05) Γ(0.3, 0.05)

Note: B(mean, sd), lnN (mean, sd), Γ(mean, sd) indicates Beta, Log Normal and Gamma distribution, re-

spectively. sd indicates standard deviation.

I follow Bayesian estimation with priors as in Table 2. Except for Germany where real interest

rate is used as risk free rate, priors are set to be identical across the countries in order to

keep estimation general. In Germany, persistence of stochastic volatility is set to have higher

mean and lower variance than the other countries, since more stable movement is expected

in risk free rate. Moreover, for the same reason, standard deviation of the volatility shock is

expected to have lower mean.

Figure 4 shows the estimated volatility for the four countries. Estimation result shows that

stochastic volatility (dash line) increased with real interest rate (solid line) for peripheral

nations during the debt crisis, especially from the end of 2011 to July 2012. Portugal shows

the highest increase in the volatility. In April 2008, volatility was as low as -1.168 in log scale.

However, at the peak of the debt crisis in August 2011, the volatility increased up to 0.878.

That is, with the same one standard deviation of level shock, real interest rate increased by

0.3% in 2008 whereas it rises by 2.4% in 2011. Time varying volatility shock can contribute

to 8 times larger movement in real interest rate in Portugal during the debt crisis.

Italy also shows sharp increase in volatility during the debt crisis. In March 2009, volatility

was as low as -1.163 in log scale, which means that there was 0.3% increase in real interest

rate when level shock hits. However, starting from the second half of 2009, volatility started

to increase continuously, hitting -0.51 in March 2011. This shows that volatility doubled two

years to 0.6% in annualized interest rate terms. Volatility kept rising until November 2011,

which peaked at -0.06 in log scale, which is 0.942 when exponentiated. This is more than

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Figure 4: Real interest rate (solid line, left scale) and estimated volatility (dash

line, right scale).

three times bigger impact on the real interest rate compared to 2009. The volatility dropped

to early 2011 level by the end of 2012, and decreased further to the lowest level in the sample

period.

In case of Spain, volatility started to increase earlier than other peripheral countries. In March

2010, volatility rose to -0.415 in log scale, which is 0.66 when exponentiated. Compared to

August 2009 where volatility was as low as -1.18 in log, or 0.307, the same level shock can

increase the real interest shock twice as more in 2010 compare to 2009. This high volatility

remained until the first half of 2012, peaking at -0.28 in June 2012 and quickly declining

afterwards.

On the other hand, German real interest rate level went down during the same period,

showing the lowest volatility over the sample. In contrast to the peripheral countries, German

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Table 3: Posterior Summary Statistics

Germany Spain Portugal Italy

Median 5% 95% Median 5% 95% Median 5% 95% Median 5% 95%

ρ 0.9708 0.9526 0.9858 0.9419 0.9252 0.9567 0.9379 0.9200 0.9543 0.9143 0.8886 0.9369

ρσ 0.9496 0.9019 0.9791 0.9021 0.8292 0.9536 0.9356 0.8833 0.9677 0.9156 0.8444 0.9595

σ -1.2547 -1.6237 -1.0087 -1.1642 -1.4599 -0.9458 -1.0354 -1.4753 -0.6910 -1.1092 -1.4387 -0.8799

η 0.1317 0.0863 0.1924 0.1806 0.1160 0.2676 0.2208 0.1566 0.3013 0.1634 0.1082 0.2490

Note: Germany is used as risk free rate. Other countries are shown for country spread = real interest rate -

risk free rate.

volatility peaked in July 2009 at -0.85 in log scale. As the volatility in other countries goes up,

Germany observed continued drop in volatility that hit its bottom in May 2012 at -1.3855.

This means that impact of volatility is nearly halved compared to the summer of 2009. For

one standard deviation shock, real interest rate in Germany increased by 0.43% in 2009 but

only 0.25% in 2012 for annualized real interest rates. This is a sharp contrast to the other

peripheral countries, whose country spread more than doubled during the same time period.

The main result of estimation is summarized in Table 3. I report Median and 5%, 95%

quantile in posterior distribution. Parameters are in general estimated with tight bands

around the median. Median of volatility level (σ) shows that Portugal has the highest mean

volatility and Germany the lowest, which is intuitive as seen in the figure above. Standard

deviation of volatility shock (η) shows qualitatively similar result, preserving the ranking in

the mean volatility. Autoregressive coefficients for both level shock (ρ) and volatility shock

(ρσ) shows more persistence for Germany compared to peripheral countries.

In the following section, I build a small open economy model that can analyze the impact of

fiscal policy when there is volatility shock. Based on the estimation result presented above,

I calibrate in Section 4 to peripheral European economies, especially focusing on Portugal.

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3 The Model

I build a small open economy model where sovereign government takes the real interest rate

as given. Market is incomplete, so there is only risk free bond that can be traded. I follow

closely to the standard setting of small open economy DSGE model. There are households,

firms and government.

3.1 Household

I denote the state of economy at each period t as st, with the history of states up to period

t as st = (s0, s1, . . . , st). Flow utility of household for any given history st is given as power-

separable form, with consumption and leisure.

u(ct(st), lt(s

t)) =ct(s

t)1−ν

1− ν− ψlt(s

t)1+η

1 + η(6)

where ct(st) denotes the consumption and lt(s

t) is labor supply.

There are three different marginal taxes, or wedges, that are given to the households: tax

on consumption τc,t(st), labor income τl,t(s

t) and capital income τk,t(st). Taking as given

the tax rates, wages wt(st), rental rates Rt(s

t), price of bonds qt(st) as well as the lump

sum government transfer Tt(st), household decide on consumption goods purchase ct(s

t),

investment in physical capital by xt(st), international bond issuance dt+1(s

t) and labor supply

lt(st). Then, the household budget constraint is:

(1 + τc,t(st))ct(s

t) + xt(st) + dt(s

t−1) + Φd(dt+1(st))

= (1− τl,t(st))wt(st)lt(st) + (1− τk,t(st))Rt(st)kt(s

t−1) + qt(st)dt+1(s

t) + Tt(st) (7)

where the physical capital depreciates with rate δ and there is capital adjustment cost

Φk(kt+1, kt):

xt(st) = kt+1(s

t)− (1− δ)kt(st−1) + Φk(kt+1(st), kt(s

t−1)), xt(st) ≥ 0 (8)

Φk(kt+1, kt) =φk2

(kt+1 − (1− δ)kt

kt− δ

)2

kt. (9)

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Notice that sign convention of budget constraint indicates positive amount of dt+1(st) as

debt. dt+1(st) is designed to represent the net borrowing of the economy, since there is

indeterminacy of private and public bond holdings. The financial asset holdings between the

government and the household will not be pinned down since the government can always

borrow from the household and return it through the lump sum transfer. Therefore, for

simplicity, I assume that only household can issue risk-free international bond dt+1(st).

Household also pays for the bond holding cost, Φd(dt+1(st)). It is a quadratic function of

(dt+1(st)− d), where d is a steady state level of international borrowing.

Φd(dt+1) =φd2

(dt+1 − d)2 (10)

Bond holding cost has been used to prevent the small open economy from displaying random

walk of real allocations. See Neumeyer and Perri (2005) for the similar application to the

small open economy and Schmitt-Grohe and Uribe (2003) for other ways to close the model.

3.2 Firms

A representative firm produces with constant returns to scale technology, renting capital and

labor from household.

yt(st) = F (kt(s

t−1), lt(st)) = kt(s

t−1)α[eXt(st)lt(st)]1−α (11)

where Xt is a labor augmenting productivity shock that follows AR(1) process: Xt =

ρXXt−1 + σXuX,t where uX,t follows i.i.d. standard normal distribution.

3.3 Government

Government need to finance a stream of government purchase gt(st) and lump sum transfer

to the household Tt(st). Linear taxation on labor, capital and consumption {τl,t, τk,t, τc,t}

together with domestic bond dgt+1 will raise government revenue. No arbitrage condition

prescribes that government faces the same risk-free bond price as the household does, so

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that household is indifferent between the domestic government bond and international bond.

Formally, government budget constraint is:

gt(st) + Tt(s

t) + dgt (st−1)

= τc,t(st)ct(s

t) + τl,t(st)wt(s

t)lt(st) + τk,t(s

t)Rt(st)kt(s

t) + qt(st)dgt+1(s

t). (12)

3.4 Competitive Equilibrium

Aggregate resource constraint of the economy is the following:

ct(st) + xt(s

t) + gt(st) + dt(s

t−1) + Φd(dt+1(st)) = F (kt(s

t−1), lt(st)) + qt(s

t)dt+1(st). (13)

Given initial conditions {d0, dg0, k0}, a competitive equilibrium is sequence of allocations

{ct(st), lt(st), kt+1(st), dt+1(s

t), dgt+1(st)}, taxes {τc,t(st), τl,t(st), τk,t(st)}, government expen-

diture and transfers {gt(st), Tt(st)}, and prices {rt(st), wt(st), R(st)} such that i) given prices,

the allocations solves household’s and firm’s problem, ii) government budget constraint is sat-

isfied, iii) resource constraint holds and iv) markets clear.

3.5 Time Varying Wedges

Tax wedges {τc,t(st), τl,t(st), τk,t(st)} and government spending {gt(st)} follows VAR(1) pro-

cess. As in Chari et al. (2007), I assume simple processes that can capture the main wedges in

the economy, which can be also interpreted as taxes. Let the vector of taxes and government

spending denoted as τt(st) = [τc,t(s

t), τl,t(st), τk,t(s

t), gt(st)]. Let ρτ ∈ R4×4 denote VAR(1)

coefficient and στ ∈ R4×4+ be variance covariance matrix. Also, let the tax wedges have steady

state mean of τ = [τc, τl, τk, g]. Then, the formal process for the tax wedges will be:

τt(st) = (I − ρτ )τ + ρττt−1(s

t−1) + exp(στ )uτ,t (14)

where uτ,t ∼iid N(0, I) is multivariate standard normal distribution.

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3.6 Interest Rate Process with Volatility Shock

In order to inspect the role of volatility shock on the real allocations and taxation, I simplify

real interest rate process into mean real interest rate r plus country spread shock εr,t.

rt = r + εr,t (15)

Country spread shock εr,t follows the same process as used in estimation section. It has

AR(1) process with level shock ur,t multiplied to the time varying volatility σt.

εr,t = ρrεr,t−1 + exp(σt)ur,t (16)

Time varying volatility process σt follows AR(1) process as well, with unconditional mean σ.

Standard deviation of the volatility shock uσ,t is given as ησ.

σt = (1− ρσ)σ + ρσσt−1 + ησuσ,t (17)

Both level and volatility shocks ur,t and uσ,t are assume to follow i.i.d. standard normal

N(0, 1), independent to each other.

4 Results

In this section, I solve the model by perturbation method and examine the model economy

in the following ways. First, I set up two economies with fixed real interest rate volatility,

high and low. By comparing the two extreme cases of volatilities, I inspect the difference in

impulse response function of tax wedges and Laffer curves. This exercise shows the possible

impact that volatility shock on the macroeconomic variables under time varying tax wedges

as well as fixed marginal tax rates.

Next, I let the volatility vary across time stochastically and inspect its impact. I compare

the effect of volatility shock on the macroeconomic variables as well as government revenues

and compare it to that of level shock. I show that impulse response function of volatility and

level shock and discuss with existing results in the literature.

15

Finally, I estimate stochastic volatility in recent European debt crisis for four countries:

Germany, Italy, Spain and Portugal. Using particle filtering and backward smoothing, I

present posterior estimation of the volatility parameters.

4.1 Model Solution

I solve the model using perturbation method at steady state. I approximate the solution up

to third order as in Fernandez-Villaverde et al. (2011a). The model solutions are derived

from the first order necessary conditions of Lagrangian program. I drop the history of states

st to simplify the notation.

uc(ct, lt) = (1 + τc,t)λt (18)

λt(qt − φd(dt+1 − d)) = βEtλt+1 (19)

ul(ct, lt) = λt(1− τl,t)(1− α)yt/lt (20)

λt(1 + Φ1k(kt+1, kt)) = βEt

[λt+1

(1− δ − Φ2

k(kt+2, kt+1) + (1− τk,t+1)αyt+1

kt+1

)](21)

where λt is Lagrangian multiplier for the household budget constraint and uc(ct, lt), ul(ct, lt)

indicate marginal flow utility of consumption and labor. Φ1k(·, ·) and Φ2

k(·, ·) denote partial

derivative of capital adjustment cost with respect to first and second argument, respectively.

I assume that international lenders are risk neutral. The bond price will be

qt =1

1 + rt+1

(22)

where rt+1 denotes the bond yield a small open economy faces.

The key to the model dynamics will be equation (19). As the real interest rate gets either

higher or volatile, household will choose to run down its debt level and reduce the consump-

tion for the next period. See Neumeyer and Perri (2005) for the level effect of real interest rate

and Fernandez-Villaverde et al. (2011a) for the volatility shock in the small open economy

business cycle.

In this paper, I focus on the interaction of tax wedges to the real interest rate shock. When

the consumption tax rate is fixed across the time periods, the Euler equation will remain the

16

same in the economy without consumption tax wedges. However, the main difference will be

in factor income tax rates that show up equations (20) and (21).

As will be shown in the following section, the distortion of labor and capital income tax will

be the highest at the peak of the Laffer curve given the constant return to scale production

technology. Government revenue will be lower with higher volatility in real interest rate for

any combination of tax rates. The mechanism can be seen through the interaction between

tax wedge and distorted marginal utility of consumption {λt}. With higher volatility of real

interest rate, consumption and investment is lower compared to the less volatile economy.

This will result in lower consumption tax base as well as lower factor income base when labor

supply is inelastic enough. Given fixed tax rate throughout the periods, the difference in

government revenue will be maximized when the low volatility economy is at its peak of the

Laffer curve, since the government revenue is pro-rata to the tax base in CRS production

function.

In order to solve for the steady state of the model, I assume that government expenditure,

tax series and transfer to the household have steady state denoted by {g, τc, τl, τk, T}. Given

these steady state values of the government as well as the steady state debt level d, allocations

{c, l, k} which are steady state consumption, labor and capital can be solved. Transfer to

household is then derived as a residual from the government budget constraint.

4.2 Comparative Statics for Volatility Shock

In this section, in order to highlight the main feature of the result, I present a simple numerical

example by comparing two economies with high and low volatilities. Since the model is solved

around the steady state, I first show the Laffer curve at the steady state without any shocks.

Then, I simulate the model under two different real interest rate volatilities, and compare

their present value Laffer curve.

Real Interest Rate Process Throughout this section, I keep the simplest real interest

rate process. I format the real interest rate as steady state level plus country spread shock,

17

which follows AR(1) process:

rt = r + εt (23)

εt = ρrεt−1 + exp(σj)ut, σj ∈ {σH , σL} (24)

where ut ∼iid N(0, 1). I choose two values of volatilities, σH for high and σL for low. I

also include AR(1) labor augmenting productivity shock Xt where Xt = ρXXt−1 + σXuX,t,

uX,t ∼iid N(0, 1). I assume the two shocks {ut, uX,t} are independent.

Calibration I calibrate the model quarterly, in line with the frequency of macro aggregates

data that will be used in estimation section. Calibration values are summarized in Table

4. With power-separable utility function, I set the inverse of elasticity of intertemporal

substitution (EIS) at 5, so that EIS is 0.2. The benchmark calibration for Frisch elasticity

is 0.02. The main purpose of low Frisch elasticity is to capture one of the most stylized

small open economy business cycle fact that with high and volatile real interest rate, output

goes down. Since Frisch elasticity of 1-5 range will result in significant increased output with

the real interest rate shock as households increase their labor supply, I keep the low Frisch

elasticity in benchmark and later show the result with Frisch elasticity of 1.

For parameters in production technology, I use the constant share of capital income of 0.38

and capital depreciation rate of 7% per annum. Both figures are from Trabandt and Uhlig

(2011), where static Laffer curve of US and EU-14 is updated.

Real interest rate is put as 2% per annum, which is close to Portugal’s real interest rate

sample mean as will be shown in Table 1 of estimation section. High volatility is chosen to

be log(0.2), which will be 10 times bigger than the low volatility log(0.02) when exponentiated

in the process.

Tax rates are set according to the average marginal effective tax rate of Portugal in 2000-

2012. Labor income tax rate is 23.43%, capital income tax rate is 31.14% and consumption

tax rate is 18.51%. Data is from European Commission where marginal effective tax rate

relevant to the macroeconomics is calculated using the tax revenue data from each country.

Marginal tax rates of other peripheral European countries used for estimation can be found

18

Table 4: Calibration

Parameter Description Value

ν Inverse of elasticity of intertemporal substitution 5

η Inverse of Frisch elasticity 50

ψ power-separable utility function parameter 1

α capital income share 0.38

δ capital depreciation (per annum) 7%

φk capital adjustment cost 5

φd bond holding cost 1

d steady state external debt 4

r steady state real interest rate (per annum) 2%

ρr autoregressive of real interest rate shock 0.95

σH volatility of real interest rate shock, high log(0.2)

σL volatility of real interest rate shock, low log(0.02)

τl mean marginal effective tax rate, labor income 23.43%

τk mean marginal effective tax rate, capital income 31.14%

τc mean marginal effective tax rate, consumption 18.51%

in Table 7 in Appendix6. Other parameters are set as in benchmark calibration, Table 4.

For parameters used in tax wedges and government spending processes, autoregressive co-

efficient is set as 0.9 and standard deviation of white noise is set as log(0.01) for all. It is

assumed that there is no correlation between any of the wedges, so that each process is a

autoregression of its own.

Real Interest Rate Impulse Response Function Based on the model solution and

calibration, I first study the impact of real interest rate shock under different volatilities,

holding every other shocks as constant at the steady state level. Figure 5 shows impact of

real interest rate volatility on government revenue as well as tax bases and borrowing decision.

6For more details regarding the methodology of calculating marginal effective tax rate, see Razin and

Tesar (1994) and Taxation Trends in the European Union (Eurostat et al., 2014).

19

Figure 5: Impulse Response Function, Real Interest Rate Shock

Note: Solid line is for high volatility and dash line is for low volatility.

Given one standard deviation shock, higher volatility economy (solid line) will have larger

increase in real interest rate on impact, whose magnitude is 10 times larger than less volatile

economy (dash line). All figures are scaled to the percentage deviation from the steady state.

On the impact of real interest rate shock, both high and low volatility economy experience

decrease in government revenue but its magnitude is bigger for more volatile economy. Dete-

riorated consumption tax base on impact and lower output in the long run work as the main

channels for reduced government revenue. As increase in real interest rate works as adverse

shock in the economy, household consumption and investment decreases. Debt level decreases

as external debt becomes more expensive and labor supply increases to smooth consumption

throughout the periods. Given low Frisch elasticity, increase in labor supply and output on

impact is transitional and marginal in its magnitude. What drives the long run transition

in output is decline of capital stock as the investment is cut with higher real interest rate.

Lower level of consumption together with lower capital stock and output level results in lower

government revenue for both economy for more than 30 quarters, whose magnitude is larger

for more volatile economy.

20

Figure 6: IRFs, Capital Income Tax

Note: Solid line is high volatility and dash line is low volatility.

Tax Wedge Impulse Response Function I first compare the impact of tax wedge shocks

under different volatilities in real interest rate7. There will be one standard deviation shock

for both tax wedges and real interest rates, holding everything else equal. Innovation on

tax wedge will be given to only one tax wedge at a time, so that difference across the tax

wedges is analyzed. The model is simulated over 100 periods, which will be 25 years based

on quarterly calibration. All figures are scaled to the percentage deviation from the steady

state as before.

In Figure 6, I show the impulse response function of the capital income tax shock. Comparing

the responses of high volatility (solid line) economy to the low volatility (dash line) one, the

direction of household decision is opposite in consumption and labor supply. When real

interest rate is volatile, household consumes less and work more in response to the adverse

shock. However, when capital tax shock is added to the stable economy, household smooth

out the negative shock over time by consuming more and working less on the impact. This

is because investment and capital stock will be reduced with increased capital tax, which

decreases output level over time. Moreover, households face higher cost in borrowing, so

7In this subsection for the Figure 6, Figure 7 and Figure 8, I solve the model up to the second order. For

all the other subsections following them are solved up to the third orders

21

Figure 7: IRFs, Labor Income Tax

Note: Solid line is high volatility and dash line is low volatility.

they run down their debt on impact. Therefore, in stable economy, the agents choose to

increase their consumption on impact in expectation of lower long run consumption level,

which is less than steady state level at the end of simulation periods.

Figure 7 shows response of the economy to the labor income tax shock. With increased labor

tax wedge, both high and low volatility economy will reduce its labor supply. However, the

difference is in its magnitude. With the same amount of increase in labor tax, stable economy

will cut back the labor supply more than the volatile economy. The reason is that when there

is large increase in the real interest rate, agents in the volatile economy need to reduce their

borrowings and cut the consumption more than those in the stable economy. Therefore, by

providing more labor supply and keeping the higher output level on the impact than the

other economy, household smooth out the negative shock.

Finally, Figure 8 shows the different reaction of the two economies from the consumption

tax wedge shock. What contrasts from the former two shocks is that households in both

economies increase their investment, whereas with capital and labor income shocks they

decreased the amount of investment. It is because capital and increased future output help

the household to smooth out their consumption over time, as borrowing and consumption

got more expensive. Labor supply decision between the volatile and stable economy shows

22

Figure 8: IRFs, Consumption Tax

Note: Solid line is high volatility and dash line is low volatility.

different direction, as the household in more volatile economy need to bump up their output

level by working more on the impact of negative shock.

Overall, the impulse response functions show that the same 1 percentage point increase in tax

wedge and one standard deviation real interest rate shock will bring very different reactions

from the households across the tax wedges as well as the volatilities in the real interest rate.

Turning to the comparison across the impulse response functions in terms of government

revenue and welfare, Table 5 shows the summarized results. Present value is calculated by

discounting from the resulting real interest rate in the impulse response functions for high

and low volatility. In the table, results for the no real interest shock and no tax shock is also

included. Impulse response functions for the no interest rate shock is in Figure 17 to 18 in

Appendix.

Inspecting the government revenue in Table 5, the present value revenue drops around 1.5%

across all tax rates. Among them, consumption tax rate records the least drop, 1.4496% in

ratios of high volatility to low volatility, as well as in absolute values. Labor and Capital

income tax follows in the scale of revenue drop, showing 1.4486% and 1.4505% respectively.

However, high government revenue also yields biggest welfare loss. For the consumption tax

23

Table 5: Present Value Government Revenue and Utility

τk τl τc r

G.Rev

σL 134.6970 134.7912 134.8744 134.5550

σH 132.7432 132.8368 132.9192 132.6020

τ 134.9035 134.9977 135.0811 .

Utility

σL 39.2724 39.2713 39.2734 39.2729

σH 38.6583 38.6571 38.6590 38.6588

τ 39.3374 39.3362 39.3384 .

Note: τx for x ∈ {k, l, c} denotes the IRFs where there were both tax rate and real interest rate shock. r

denotes the case where there is no tax shock. σL is low volatility economy and σH is the high volatility. τ

is for only tax rate shock, as in Figure 17 to 19. Here, utility is presented after affine transformation of the

utility function. That is, for flow utility u(c, l), the figures above are calculated with u(c, l) = 200∗u(c, l)+1.

increase in high volatility, welfare drops 1.5644% compared to the low volatility. In labor

and capital tax, welfare decreases by 1.5640% and 1.5637%, respectively.

Therefore, the government faces the choice between high government revenue and large wel-

fare loss when increasing the tax rates among the volatile real interest rates. Absent real

interest rate shock whose volatility is time varying, government revenue collection will be

overestimated whereas welfare loss will be underestimated. In the context of European

peripheral government that undertook austerity programs, this study tells that household

suffered more during the debt crisis for the same tax increase compared to the less volatile

periods.

Simulated Laffer Curve In this subsection, I hold the tax wedges fixed at its steady

state level and investigate the impact of volatility in government revenue by simulated Laffer

curves for each taxes. Each Laffer curve is plotted holding all other taxes at 0, and varying

24

Figure 9: Laffer Curves

Note: Laffer curves for capital income (top), labor income (center) and consumption (bottom) tax, where

simulated Laffer curves are on left column and static Laffer curves are on right column. Simulated Laffer

curves under high volatility is solid line and low volatility is dash line.

only one tax rate each time. Present value of government revenue is derived from each

simulation under different volatility, discounted by the inverse of realized gross real interest

25

Figure 10: Slope of Consumption Tax Laffer Curve

Note: Simulated Laffer curve slope (left) and static Laffer curve slope (right), where dash line on the left is

for low volatility economy and solid line is for high volatility one.

rate. Simulated Laffer curve is also compared with traditional steady state Laffer curve for

each taxation. However, the simulated Laffer curve and static Laffer curve is not directly

comparable as with third order approximation method, simulation value easily gets away

from the steady state value. Therefore, the main focus will be the difference between high

and low volatility Laffer curve and the shape-preserving characteristic of simulated Laffer

curve compared to the static one.

Details of simulation method is as follows. Since the model is solved using third order

approximation, the shocks are restricted to be 1 standard deviation as in Fernandez-Villaverde

et al. (2011a). Perturbation method is performed at the steady state rather than ergodic

mean, since the comparison to static Laffer curve is more relevant when simulated around

the steady state. Calibration values are as in Table 4 and tax grid is set from 0 to 99% with

1% steps.

In order to highlight the role of volatility in government revenue in the context of debt crisis, I

show in this paper a sample Laffer curve with burn-in period 200 and simulation time period

100, which is 25 years as the model frequency is in quarters. Details of the sample path of

simulation including the interest rate process is showed in Figure 16 in Appendix.

The most curvature of Laffer curve can be observed with capital tax in Figure 9. Both

26

Figure 11: Laffer Curves, Frisch Elasticity of 1

Note: Simulated Laffer curve for capital Income tax (left) and labor income tax (right). Laffer curves under

high volatility is in solid line and low volatility is in dash line.

simulated and static Laffer curve peaks at 64% of capital tax, which are marked by circles in

the graph. Simulated Laffer curve on the left shows that present value government revenue

is lower over all tax grids for the higher volatility real interest rate (solid line) than the lower

one (dash line). Also, observe that the difference between two economies is the largest at the

peak of the simulated Laffer curve.

Next, labor income tax Laffer curve shows monotonic increase of labor tax revenue due to

low Frisch elasticity in calibration. Simulated and static Laffer curves all peak at 99% labor

income tax rate. Qualitative characteristics remains the same as the capital tax Laffer curve,

such that the difference between the high and low volatility economy gets larger at the peak

of the Laffer curve and shape of static Laffer curve is preserved in simulated one.

Finally, consumption tax Laffer curve in the bottom of Figure 9 shows seemingly linear line

for both static and simulated ones. This is also in line with Razin and Tesar (1994) where it

is shown that there is no peak of consumption Laffer curve. However, once inspected closely,

Laffer curve for consumption tax is not entirely linear. As Figure 10 shows, the slope of each

point in the Laffer curve is not constant, lower for tax rates close to 0% and converging as

the tax rate is closer to 100%. Therefore, consumption Laffer curve also shows “Laffer hill”

for lower tax rate, albeit subtle.

27

Figure 12: IRFs, level and volatility shock

Note: Volatility shock (solid line) and level shock (dash line)

Frisch Elasticity and Laffer Curve In above result, labor income tax Laffer curve is

monotone due to the low Frisch elasticity that induces almost inelastic labor supply. There-

fore, I present Laffer curves with higher Frisch elasticity, which is set at 1. As can be seen in

Figure 11, labor income tax rate shows much variation with higher Frisch elasticity, which

peaks at 87% for both static and simulated Laffer curve. Capital income Laffer curve, on

the other hand, peaks later with higher Frisch elasticity, at 81% as opposed to 64% in the

benchmark case. The main characteristics of the static and simulated Laffer curves are pre-

served in the higher Frisch elasticity as well. Static Laffer curves for factor income taxes and

consumption Laffer curves are presented in Figure 15 in Appendix.

4.3 Effect of Volatility Shock on Government Revenue

In this subsection, I compare the level shock and volatility shock through the impulse response

function of each shock. In particular, I closely calibrate to Portugal data using the estimated

result in Section 2 and study highlight the role of volatility shock. I abstract from volatility

shock in risk free rate and only put stochastic volatility in country shock. Parameters are

calibrated to median value in Table 3 for Portugal: autoregressive coefficient for level shock ρ

28

is 0.9379 and for volatility shock ρσ is 0.9356. Mean level of volatility shock σ is -1.0354 and

standard deviation of volatility shock is 0.2208. Mean interest is set at 2.18% per annum,

which is the mean real interest rate level in Table 1. All other parameters are kept as in

Table 4. Impulse response functions are in percent deviation from the steady state level.

Volatility shock impacts the household decision through the precautionary behavior, without

increasing the real interest rate. Compared to the level shock, household run down the

debt more, cut back the consumption and work more. Consumption in both level shock

and volatility shock decreases on the impact and subsides to the new level. In all household

decisions, level shock shows smoother transition compared to the volatility shock. In terms of

capital accumulation, volatility shock leads the household to invest more on the impact so that

capital stock goes up more than the level shock in the short run. Due to the precautionary

motive, capital stock shows hump-shaped transition to the new higher level than the steady

state.

As a result of household decision, government revenue decreases on the impact but increase

in the long run due to higher capital stock and output level. In the short run, consumption

level drops so that government revenue raised from the consumption decreases, lowering the

aggregate government revenue. However, since household borrows less, invests more and

provides more labor supply, output level goes up and this increases the government revenue

from the factor income taxes.

Compared to the previous studies, this model has smooth capital stock transition but the

investment decision fluctuates over time. This is mainly because the capital adjustment cost

is put on the stock of capital rather than the investment decision. Nevertheless, the other

variables such as consumption, labor supply and borrowings show consistent result as in the

previous studies8.

8See Fernandez-Villaverde et al. (2011a) Figure 5 for their result of impulse response function comparing

level and volatility shocks.

29

5 Conclusion

This paper studied the interaction of tax wedges and volatility shock of real interest rate in a

small open economy. Based on European data, I first showed that there is significant degree

of time variance in volatility of government bond yields. Then, in the model I showed that

the same increase in tax wedge will give different response to the economy when combined

with real interest rate shock. Also, higher volatility will lower down the peak of Laffer

curve, preserving shape of the curve. After comparing static volatilities, I compared level

and volatility shock and showed that their impact on real variables especially in investment

decision.

In this paper, I have focused on the mechanisms and dynamics of the model, with calibrated

figures outside of the model. For the future research, the quantitative model proposed in

this paper can be served as a basic framework for the business cycle accounting in a small

open economy. This will enable the decomposition of the wedges that impacted European

countries during the debt crisis.

Finally, this paper assumes that real interest rate is subject to exogenous shocks with a

stochastic process, not an equilibrium object. It will be interesting to study the interaction

of government’s decision on taxation and real interest rate that the economy faces. However,

it is beyond the scope of this paper to incorporate the feedback effect of bond price to the

fiscal policy, and I leave it to the future research.

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31

A Appendix

Figure 13: Real Total General Government Revenue Index

Figure 14: Real GDP Index

Note: Real total general government revenue and real GDPs are from Eurostat. Index is calculated by the

author.

32

Steady State Laffer Curve Steady state Laffer curve is characterized by the following

equations.ψlη+1

(1− α)y=

1− τl1 + τc

c−ν (25)

y = kαl1−α (26)

1 = β[1− δ + (1− τk)αy/k] (27)

c+ δk + g = y + (1/(1 + r)− 1)d (28)

Sample Path of Simulation In order to understand how the model economy behaves

under different volatilities of level shock, I compare two sample economies. One economy is

given with high volatility σH and the other is given with low volatility, σL. Illustration of

the sample path is given in Figure 16.

Impulse Response Function I present the impulse response function when there is only

tax rate shock in Figures 17 to 19. These figures will complement Section 4.2. Also, I present

the present value of government revenue and utility for each impulse response function in

Table 5.

33

Figure 15: Laffer Curve, Frisch Elasticity of 1

Note: Laffer curves of different taxation in simulated model with varying volatility (left column) and steady

state (right column). On left column, solid line is under high volatility and dash line is under low volatility.

Taxes are capital income (top), labor income (center) and consumption (bottom).

34

Figure 16: Real Allocation, simulation sample.

Note: Simulation sample path of high volatility (solid line) and low volatility (dash line).

35

Figure 17: IRFs, Capital Income Tax

Figure 18: IRFs, Labor Income Tax

Figure 19: IRFs, Consumption Tax

Note: Simulation sample path of high volatility (solid line) and low volatility (dash line).

36

B Data Appendix

B.1 Real Interest Rate

Spain, Portugal and Italy data for country spread and Germany for risk free interest rate.

Nominal interest rates are from government bond yields, 3 year maturity. Data sourced from

each country’s central bank. Harmonized Consumer Price Index (overall index) for each

country is from Bundes Bank. Real interest rate = Yield - HCPI (percent per annum). Risk

free rate is German real interest rate. Country spread = Real interest rate - Risk free rate.

Monthly data from January 2000 to April 2015.

B.2 National Accounts

Consumption: P31 S14 Final consumption expenditure of households. Investment: P5 Gross

capital formation. Net export: P6 Exports of goods and services - P7 Imports of goods and

services. Output: B1GM Gross domestic product at market prices.

Quarterly real national account data from Eurostat, namq gdp k. From 2000Q1 to 2014Q1.

Million Euro Chain linked volume 2005. Data is downloaded for not seasonally adjusted. All

items are seasonally adjusted by X-13ARIMA-SEATS by US Census Bureau and then log

scaled. All series are hp-filtered with λ = 1600 for the moments calculation. nx/y is a mean

of (seasonally adjusted, not hp-filtered) net export to output ratio.

B.3 Taxation

Taxation data from European Commission “Taxation Trends in the European Union” 2014

edition. In Annex A, implicit tax rate of capital, consumption and labor. Annual data.

Sample period 2000 to 2012. Greece does not have implicit capital tax rate. Government

Expenditure to GDP data from Eurostat. Data is annual. [gov a main]. Sample period is

2000 to 2013.

37

Table 6: National Account Second Moment

EL ES IT PT

σy 0.02 0.01 0.01 0.01

σc/σy 1.43 1.20 0.69 1.08

σi/σy 5.80 3.65 3.03 3.61

nx/y -0.10 -0.02 0.01 -0.07

σnx 0.15 0.37 1.15 0.16

corr(nx, y) 0.69 0.77 0.40 0.22

Table 7: Tax Data Summary Statistics

country τL τC τK g/y

Mean

Greece 33.22 15.92 . 48.17

Italy 42.11 17.62 30.89 48.58

Portugal 23.43 18.51 31.14 45.92

Spain 32.26 15.14 30.45 41.35

SD

Greece 1.952 0.608 . 3.546

Italy 0.609 0.414 3.418 1.659

Portugal 1.062 0.863 2.490 2.889

Spain 0.894 1.228 5.470 3.685

Note: SD indicates standard deviation. Capital taxation for Greece not available in the data.

38